where to start on putting a new edge on a cheap axe ?

[Daz]

so you go buy a cheap axe and it's not upto the job
where and how would you go about putting a good edge on it ?

Daz

[Adam Savage]

I found with cheap axes, the steel is normally quite soft. If you are careful, without holding it on too long and scorching the steel, try using a very, very light going over with a grinder, then put the final edge on with a stone. Worked for my hatchet and multi axe.

[Humakt]

As Crazysaint says.
Take the worst off with a grinder, being careful not to let any one area get too hot.
Then out with the grinding stones or whatever your preferred sharpening method is (personally I prefer 600 and 1200 grit wet and dry - I get a much better finish with it and the materials are dirt cheap as well. But use whatever you prefer to get the job done, that's the most important thing). I tend to wrap the wet and dry around a block of wood, spit on it to get it wet, and then hold the axe firmly on something flat (e.g. work top), with the cutting edge hanging over the edge, and then use a circular motion to bring it up nice and sharp. 30 or 40 swirls one way (as if I actually count) and then 30 or 40 swirls the other. Being careful to hold your polishing stones/wet and dry at the angle you want the final edge to be.
Be careful with cheap axes. A lot of the new cheap axes (e.g. the Argos ones) are made of very poor steel, and don't hold a good edge for very long. Conversely, a lot of old cheap axes you get from Boot Fairs could well be made of much superior steel (though there are, of course, some real stinkers as well) that can hold a good edge for a long time. These cheap axes (from Boot Fairs) are the best buys - some modern, mass-produced tools are made from poor steel since the manufacturers tend to assume the buyer wants them for a single project and that's how long they are made to last. They know that craftsmen are happy to spend more money on decent tools since they are an investment. That's not to say there's not a place for these new cheap tools, because there is - as a place to learn the basics of tool care and sharpening. So that's what I would do if I was you - use your cheap axe as a means to learn such things as sharpening techniques and so on. It'll more than pay for itself in that respect and will definitely be a worthwhile buy.